ANALYSIS part of the Exercise Set 4

1&2. Reading the Boston data from the MASS package and exploring it

library(MASS)
data("Boston")
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506  14

This dataset contains information collected by the U.S Census Service concerning housing in the area of Boston Mass. It was obtained from the StatLib archive (http://lib.stat.cmu.edu/datasets/boston), and has been used extensively throughout the literature to benchmark algorithms. However, these comparisons were primarily done outside of Delve and are thus somewhat suspect. The dataset is small in size with only 506 cases.The data was originally published by Harrison, D. and Rubinfeld, D.L. `Hedonic prices and the demand for clean air’, J. Environ. Economics & Management, vol.5, 81-102, 1978.

3. Variables of Boston data.

There are 14 attributes in each case of the dataset. They are: CRIM - per capita crime rate by town; ZN - proportion of residential land zoned for lots over 25,000 sq.ft.; INDUS - proportion of non-retail business acres per town; CHAS - Charles River dummy variable (1 if tract bounds river; 0 otherwise); NOX - nitric oxides concentration (parts per 10 million); RM - average number of rooms per dwelling; AGE - proportion of owner-occupied units built prior to 1940; DIS - weighted distances to five Boston employment centres; RAD - index of accessibility to radial highways; TAX - full-value property-tax rate per $10,000; PTRATIO - pupil-teacher ratio by town; B - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town; LSTAT - % lower status of the population; MEDV - Median value of owner-occupied homes in $1000’s.

Let’ see the matrix of Boston variables.

pairs(Boston)

summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

Let’s see correlations between variables in the dataset.

library(dplyr)
## 
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
## 
##     select
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(corrplot)
## corrplot 0.84 loaded
cor_matrix<-cor(Boston) %>% round(digits = 2)
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)

Interpretation: we can see that some variables correlate greatly with ech other, for example, INDUS, NOX and AGE (positive correlation). INDUS and NOX correlate negatively. COrrelation plot is very informative about the relationships of the variables.

4.Standardization the dataset using scale() and summaries of the scaled data.

boston_scaled <- scale(Boston)
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
class(boston_scaled)
## [1] "matrix"
boston_scaled <- as.data.frame(boston_scaled)
class(boston_scaled)
## [1] "data.frame"

As we can see now, mean of each variable is set to zero.We also converted boston_scale object from matrix to the dateframe using R functions and verified the class of the object.

Now we will create a categorical variable of the crime rate from the scaled crime rate using the quantiles as the break points in the categorical variable. We need to drop the old crime rate variable from the dataset, too.

Let’s have a look at the variable “crime”.

summary(boston_scaled$crim)
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.419367 -0.410563 -0.390280  0.000000  0.007389  9.924110

Now we make a quantile vector of it and check it.

bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610

Now we make a categorical variable out of variable “crime” with 4 labels.Then we check frequences by table().

crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127

Next step is to remove original crim from the dataset and add the new categorical value to scaled data. We will also check it afterwards.

boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)
table(boston_scaled$crime)
## 
##      low  med_low med_high     high 
##      127      126      126      127

Now we will create a train dataset(80% of the dataset) and a test dataset from Boston.

n <- nrow(boston_scaled)
ind <- sample(n,  size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]

5.Linear discriminant analysis

Let’s run linear discriminant analysis.

lda.fit <- lda(crime ~ ., data = train)
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2549505 0.2450495 0.2524752 0.2475248 
## 
## Group means:
##                  zn      indus        chas        nox         rm
## low       1.0640380 -0.9036420 -0.15765625 -0.8819051  0.4110106
## med_low  -0.1712920 -0.2584181 -0.03371693 -0.5239687 -0.1556059
## med_high -0.3939194  0.2175024  0.22945822  0.4046783  0.1561290
## high     -0.4872402  1.0171519 -0.03610305  1.1087118 -0.4566418
##                 age        dis        rad        tax      ptratio
## low      -0.9069034  0.8733585 -0.6975420 -0.7332905 -0.468721678
## med_low  -0.2739818  0.2647897 -0.5468458 -0.4575606  0.004675854
## med_high  0.4274056 -0.3920281 -0.3817409 -0.2771944 -0.289661896
## high      0.8238024 -0.8676900  1.6377820  1.5138081  0.780373633
##                black        lstat        medv
## low       0.37954843 -0.758971107  0.50070737
## med_low   0.31733792 -0.113305822 -0.01619727
## med_high  0.04488567 -0.003202208  0.20301974
## high     -0.77278971  0.901999899 -0.63844078
## 
## Coefficients of linear discriminants:
##                   LD1         LD2         LD3
## zn       0.0880161075  0.88509974 -0.88291726
## indus    0.0115844326 -0.24837392  0.13785851
## chas    -0.0930756421 -0.07843240  0.09532990
## nox      0.4782700500 -0.62191412 -1.45600112
## rm      -0.0928351313 -0.11964937 -0.19966134
## age      0.3040160398 -0.33863490 -0.06823650
## dis      0.0137669211 -0.50007345  0.18047576
## rad      2.9795691185  1.05506386 -0.06350704
## tax      0.0001341613 -0.20975161  0.69074677
## ptratio  0.1311160608  0.01648687 -0.17549334
## black   -0.1285281124  0.01862796  0.11207535
## lstat    0.2386136476 -0.28649024  0.36368632
## medv     0.2470427036 -0.41113274 -0.16840048
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9403 0.0439 0.0158

Now we will plot the results of lda.

lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)

6. classes prediction with the LDA model on the test data. Cross-tabulation of the results.

correct_classes <- test$crime
test <- dplyr::select(test, -crime)
lda.pred <- predict(lda.fit, newdata = test)
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       12      11        1    0
##   med_low    9      17        1    0
##   med_high   1      13       10    0
##   high       0       0        0   27

7. Reload of Boston and k-means algorithm.

library(MASS)
data("Boston")
dist_eu <- dist(scale(Boston))
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970
dist_man <- dist(scale(Boston), method = "manhattan")
summary(dist_man)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.2662  8.4832 12.6090 13.5488 17.7568 48.8618

Now we do k-means clustering and plot it!

km <-kmeans(Boston, centers = 3)
pairs(Boston, col = km$cluster)

Next step is to investigate k number.

library(ggplot2)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
qplot(x = 1:k_max, y = twcss, geom = 'line')

km <-kmeans(Boston, centers = 2)
pairs(Boston, col = km$cluster)

Interpretation: The optimal number of clusters is when the value of total WCSS changes radically. In this case, two clusters would seem optimal.SO we reran k-means with two clusters.

Super-Bonus.

model_predictors <- dplyr::select(train, -crime)
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')

Nice plot!